Local density distribution. The RDF function suggests that the higher temperature dataset D2 falls into a gas-like social phase. On the other hand, the lower temperature datasets D1A-B appear to be in the liquid-vapor coexistence-like social phase. To distinguish these two different social phases, we calculated the local density distribution (LDD) [1]}, \(p(\rho )\) , that measures the spatial variation of the local number density of particles (see SM Section B3). In the coexistence phase, \(p(\rho )\)  is bipeaked [2]} because the particles are inhomogeneously distributed in space where the low and high-density peaks represent vapor and liquid phases, respectively. In contrast, \(p(\rho )\)  is single-peaked around the average density, \(\bar{\rho }\) , for the gas phase. Figure 3a-c, plot \(p(\rho )\)  for datasets D1-D2, showing that \(p(\rho )\)  is mono-peaked for the high-temperature dataset (D2). In contrast, we find that \(p(\rho )\)  in D1 agrees well with a bi-gamma distribution, with a vapor density peak around \(\rho _v=0.2\bar{\rho }\sim 0.4\bar{\rho }\) , and a liquid density peak around \(\rho _l=1.05\bar{\rho }\sim 1.20\bar{\rho }\) . This result suggests a liquid-vapor coexistence phase in D1 and a gas phase in D2. While existing models suggest a similar liquid-vapor phase transition [2]}, these transitions are often coupled with a change of orientational order. We measured the magnetization order parameter \(m_0 \equiv \frac{1}{N}\left|\left<\left(\sum _{i=1}^{N}\cos \theta _i,\sum _{i=1}^{N}\sin \theta _i \right) \right> \right|\) , for the dataset D1, finding \(m_0 \approx 0.001\)  and \(0.002\)  for D1A and D1B respectively. This finding suggests that unlike the existing models [4]}, [2]} the liquid-vapor phase transition does not accompany by orientational order changes, partly due to the mixture of ferro- and antiferromagnetic orders at different distances.
